The first results from the Tenth Cambridge (10C) Survey of Radio Sources, carried out using the AMI Large Array (LA) at an observing frequency of 15.7 GHz, are presented. The survey fields cover an area of approximately 27 sq. degrees to a flux-density completeness of 1 mJy. Results for some deeper areas, covering approximately 12 sq. degrees, wholly contained within the total areas and complete to 0.5 mJy, are also presented. The completeness for both areas is estimated to be at least 93 per cent. The source catalogue contains 1897 entries and is available at www.mrao.cam.ac.uk/surveys/10C. It has been combined with that of the 9C Survey to calculate the 15.7-GHz source counts. A broken power law is found to provide a good parameterisation of the differential count between 0.5 mJy and 1 Jy. The measured count has been compared to that predicted by de Zotti et al. (2005). The model displays good agreement with the data at the highest flux densities but under-predicts the integrated count between 0.5 mJy and 1 Jy by about 30 per cent. Entries from the source catalogue have been matched to those contained in the catalogues of NVSS and FIRST (both of which have observing frequencies of 1.4 GHz). This matching provides evidence for a shift in the typical 1.4-to-15.7-GHz spectral index of the 15.7-GHz-selected source population with decreasing flux density towards sub-mJy levels - the spectra tend to become less steep. Automated methods for detecting extended sources have been applied to the data; approximately 5 per cent of the sources are found to be extended relative to the LA synthesised beam of approximately 30 arcsec. Investigations using higher-resolution data showed that most of the genuinely extended sources at 16 GHz are classical doubles, although some nearby galaxies and twin-jet sources were also identified.
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